# Equivalent fractions

Equivalent fractions are fractions that represent the same value. Examples include , , and , which all represent the same part of a whole. This can be depicted using fraction bars:

Even though each of the fraction bars is broken up into a different number of parts, each of the fraction bars represents the same fraction.

## How to find equivalent fractions

Any given fraction has an infinite number of equivalent fractions. We can find equivalent fractions by multiplying or dividing both the numerator and denominator of a fraction by the same number; this is essentially a fractional representation of the number 1. Since multiplying or dividing any number by 1 results in the same number, multiplying any fraction by a fractional representation of the number 1 results in an equivalent fraction.

### How to find equivalent fractions by multiplying

To find an equivalent fraction, multiply the numerator and the denominator of the fraction by the same number.

Example

Find three equivalent fractions for .

1.

2.

3.

If we were to divide each of the fractions above, we would find that they all equal 0.3.

### How to find equivalent fractions by dividing

To find an equivalent fraction, divide the numerator and denominator of the fraction by the same number, making sure that the resulting numerator and denominator will be a whole number. If it is not possible to divide the starting fraction by some number such that the resulting fraction is made up of whole numbers, use the multiplication method instead.

Examples

Find equivalent fractions for the following fractions.

1. :

2. :

3. :

Notice that for this example, we cannot divide by any number other than 1 and still have a whole number in both the numerator and denominator. In this case, we wouldn't use division, but instead could multiply both numerator and denominator by the same number to get an equivalent fraction. Since, all of the examples shown above are equivalent fractions of , we can use any of the values:

## How to test if fractions are equivalent

There are a few different ways we can test whether two or more given fractions are equivalent.

### Convert the fractions

Convert the fractions such that they have the same denominator. Once converted, if the numerators are the same, the fractions are equivalent. Otherwise, the fractions are not equivalent.

Examples

1.:

Both fractions can be converted to have a denominator of 24:

Thus, they are equivalent fractions.

2.:

Both fractions can be converted to have a denominator of 24:

Since the numerators are not the same, the fractions are not equivalent.

### Cross multiply the fractions

Another way to test if two fractions are equivalent is to cross multiply them. If the products of the cross multiplication are equal, then the fractions are equivalent.

Example

1.:

Cross multiplying the fractions yields:

Thus, the fractions are equivalent.

2.:

Cross multiplying the fractions yields:

Since the products are not equal, the fractions are not equivalent.

### Convert the fractions to decimals

Another method for determining if fractions are equivalent is to simply use a calculator, long division, or some other method to convert the given fractions to decimals. If the decimals are equal, the fractions are equivalent.

## Adding and subtracting fractions

Equivalent fractions are used when adding and subtracting fractions. In order to add or subtract a fraction, the fractions involved must be like fractions. If they are unlike fractions, then the unlike fractions must be converted into equivalent fractions that share the same denominator in order to be added or subtracted.

Examples

Convert the fractions in the following problems to equivalent fractions such that the fractions all share the same denominator, then add or subtract the fractions.

1. :

The least common multiple of 2 and 5 is 10, so we will convert the above fractions to equivalent fractions that have a denominator of 10 in order to add them.

2. :

The least common multiple of 7 and 21 is 21, so we only need to convert the first fraction to an equivalent fraction.